How do you like your contemporary art? A quick hit of juicy mischief, a larky take on mortality, binful of bluebottles, pocketful of glitter, everything you never wanted to know and more about the artist's entrails? Right then, give Anselm Kiefer a very wide berth - because, as the show about to open at White Cube, London, will confirm, he doesn't do droll, he does the big embarrassing stuff, the stuff that matters: the epic slaughters of the world, the incineration of the planet, apocalypse then, apocalypse often; the fragile endurance of the sacred amid the cauterised ruins of the earth.

But lately, the undertaker of history has turned gardener. From deep beneath the loam of memory heaped over the canvas, Kiefer's vast, rutted wastelands have germinated brilliant resurrections: pastel blooms, spikes of verdure sprouting irrepressibly through the skin of a hard-baked earth-rind; or peachy-pink poppies trembling atop spindly black stalks that climb gawkily from bituminous slag. So, notwithstanding the massacre du jour served up with the cornflakes in our daily newsprint, for Kiefer in his current redemptive temper, hope really does spring eternal.

He must be the most un-hip artist ever shown by White Cube, the very Sanhedrin of cool; barely a contemporary painter-sculptor at all, if the range framing the contemporary goes from coyly self-effacing minimalism to gaudy showboating. What's more, Kiefer does earthspace, not cyberspace. No Luddite, he none the less has let it be known that what he dislikes most about computers is the indiscriminate quality of their memory, a universe of data held simultaneously, accessible at the click of a mouse, permanently available and impervious to either natural attrition or poetic distortion. Since nothing may be digitally forgotten, nothing may be truly recalled.

Much of Kiefer's art represents a resistance to this inhuman virtualisation of memory; its lazy democracy of significance, its translation into weightless impressions. The opposing pole from that alt/delete disposability is to make history obstinately material, laid down in dense, sedimentary deposits that demand patient, rugged excavation. Kiefer's work burrows away at time, and what it exposes also makes visible the painful toil of the dig, skinned knuckles, barked shins and all.

For a German born amid the slaughterhouses of 1945, booting up could never be glibly electronic. Kiefer became famous in the 1970s and 80s for his frontal engagements with the totems of German history: blood- spattered trails befouling the deep Teutonic woods (his name means fir tree) from which the national culture had been proverbially rough-hewn; torch-lit timbered pantheons within which heroes and anti-heroes lay provisionally interred.

By the lights of the transatlantic avant garde, Kiefer did absolutely everything wrong. The choices were clear. Art either had to be hard-edged in its irreducibly angular minimalism, like Donald Judd's stacked boxes, which drew their ominous power from being nothing other than what they were; or else it had to be ecstatically collapsed into the raw and rowdy universe of signs: op'n'pop, flags and soup tins, one long cackle at art's valetudinarian pretence to hold the moral high ground. To grab our attention amid the modern clamour, art needed to drop the churchiness (especially abstract churchiness), and get out from under all those centuries of pompous sententiousness and obscure story- telling. All painting could be was flat-out play (with the emphasis on flat).

Wrestling with his Teutonic demons - and keeping close company with the likes of Grünewald, Altdorfer and Caspar David Friedrich - Kiefer could scarcely comprehend, much less identify with, the case for painterly amnesia, nor with the posturing for lightness and shallowness (he has never been much of a tease). Stubbornly, his art was always hewed to spatial depth and moral weight, so his landscapes take anti-flatness about as far as it can possibly go, opening immense vistas behind the picture, carved furrows on the surface stretching away to remote distance.

He does not do this innocently, of course. The practice of perspective, invented to imagine a bucolic world where pastoral fancies were enacted in a neverland of happy radiance, is recycled in Kiefer's landscapes to exterminate the fantasy. Kiefer's skies are often black, streaked with the phosphoric licks of a descending firestorm, and what vanishes at the vanishing point are the balmy consolations of rusticity. Bye-bye Hay Wain, hello the Somme.

Kiefer also needs immensity in order to frame the ancestral epics of life and death which for him remain art's proper quarry, and which sometimes extend beyond that far horizon into the infinite metaphysical space of the beckoning cosmos, where they interrupt the emptiness with mapped constellations. Events - scriptural, mythic, poetic, historic - are transfigured into written words on the painting, because, for Kiefer, words sanctify the events and figures to which they refer, rather than demystify them.

His seriousness about words, as weighty as the lead from which he forges his books, also puts him at odds with the ironic mode of quotation that has long defined modern art. Instead of mimicking the industrially reproduced comic signage of the mass market, Kiefer marks his pictures with the spidery inscriptions of his own hand, the moving finger quoting, inter alia: Isaiah, Paul Celan, Aeschylus. Mirabile dictu. Wordiness for Kiefer is painterliness. The library and the gallery, the book and the frame inseparable, even interchangeable, in his monumental archive of human memory. Not since Picasso's Guernica have pictures demanded so urgently that we studiously reflect and recollect in their presence.

Which may make Kiefer's new work sound like homework (to be severely marked by the forbidding Herr Professor Uber-Bombast). Nah, thanks all the same, you're thinking, would rather do a day with Damien and Trace. But advance preparation in the Iliad, the Kabbalah, not to mention higher scriptural exegesis, is really not the price of admission. For visual drama that (I guarantee) will haunt your dreams, there's no one alive to beat Anselm Kiefer. This is because, along with being a philosopher-poet, he also happens to be a craftsman of phenomenal power and versatility.

For some time, he has been experimenting with work that crosses the boundaries separating not just art and literature but painting and sculpture. Sometimes (as in the breathtaking Merkaba), a gesamtkunstwerk (total artwork) brings together free-standing sculptural elements in stone or lead in continuum with a vertical painted surface, the one acting as a terrestrial transport to the celestial apparition of the other. Some of the paintings on display at White Cube sustain this working method by setting a clump of thorn bushes before an ashy grey winterscape that speaks (much less ponderously than this makes it sound) of chill death and resurrection. But other paintings - especially in the triptych of confounding masterpieces that, alas, will be travelling to the Gallery of New South Wales in Sydney - have incorporated into the grittily loaded texture of the canvas itself a seething bed of organic (and occasionally inorganic) matter, so that the surface becomes akin to a yeasty humus; alive with golden flecks of straw and hay, twigs, whole branches which poke through the impasto. Another denial of the modernist dogma that authentic paint should never dare to present as anything other than itself.

Kiefer's paint is forthrightly the crusty medium of generation - the baked clay that develops the cracks and fissures from which vegetable life burgeons forth. Even when the paint is, in fact, just that, it is made to clot and coagulate, puddle and pond, or rise in frozen crests as if it were the volcanic material of primordial genesis.

Kiefer's painting, then, is not a representation of some feature of creation so much as a re-enactment of it. And if this sounds a mite up itself, well indeed it is, and none the worse for it. Even if you care not a toss for the esoterica, the richness of classical allusion (such as the catastrophic landscape of the fall of Troy, scarred with explosions of carbon and cobalt, and transmitted via a telephonic connection from Greek peak to peak in mimicry of Agamemnon's beacon signals to faithless Clytemnestra), you can still happily envelop yourself in the blanket of colour and line that fill every centimetre of Kiefer's pictures.

Dazzling, nostalgically psychedelic shots of colour. Beneath the verse from Isaiah that speaks of heavenly mercy, "Rorate coeli desuper et nubes pluant" (Drop down ye dew and let the clouds rain upon the just), Kiefer has planted a field of blazing, flamingo-tinted poppies. But the mercy is not unqualified; the flowers are marshalled along perspectival lines all the way to a horizon that is built from raised skeins of greenish-black paint, the corrupted hues of chemical pollution. (Evidently we're not in Monet's picnic country of Les Coquelicots.) Kiefer's poppies with their black faces can be read interchangeably as columns of warriors or the floral memorials of their fiery entombment. And the petals of the middle distance suggest the flares of combat as much as a field of flowers.

The most startlingly florid of the pictures travels from a paradise garden at its base, with the caked terracotta blossoming in arabesques of brilliant violet, pink and vermilion splashes that coil through the more furrowed landscape. Above it are more verses from Isaiah that open the Palm Sunday liturgy: "Aperiat terra et germinat Salvatorem" (Let the earth open and bring forth a Saviour). But Kiefer being Kiefer, there needs be trouble in paradise, so that along the serpentine line of beauty lurks the form of a skeletal snake, its vertebra constructed from a string of terracotta beads suspended on spinal wire poking from the picture surface. Good and evil, vitality and mortality, thus literally hang sinuously in the balance, it being deliberately unclear whether the serpent is safely fossilised within the sprouting clay, or has shed its casing the better to writhe into freshly devilish incarnation.

This dialogue between martyrdom and resurrection continues into the deeply stirring Palm Sunday installation: 18 glazed cabinets that house vertically mounted branches of vegetation (mangroves, sunflower pods and dracenae as well as palms themselves). Stacked atop each other, the cabinets seem not so much vitrines from some botanical museum as the opened pages of a herbiary, Kiefer the tree-man knowingly playing with the conceit of a super-folio, interleaved with sacred revelations. The branches are coated with a thin skin of plaster or white paint so that, at first sight, they seem bleached of life, sapless and forlornly skeletal. But the newly (if tentatively) optimistic Kiefer wants us to register Palm Sunday as a true triumph; the entry to Jerusalem inaugurating the events that lead not just to the Passion but to the Resurrection. Kiefer also knows that, in both pagan and early Christian iconography, the palm with its sword-like branches was known as an immortal tree, which never actually perished but constantly regenerated, a new sheath of fronds budding from the site of a fallen limb. The very earliest representations of the cross in the Coptic church thus took the form of the living palm. Kiefer has also contrived to display another palm as if it were the feathers of some avian or even angelic wing; a doubly miraculous apparition which, as outsize quill, writes its own revelatory gospel behind the veiling glass.

Like the dirty fields of death sewn with floral brilliance, Kiefer's phantom tree limbs enact a parable of the intertwined fate of nature and humanity. For the erect branches lie or stand against flesh-coloured beds of sand, that in their gracefully voluptuous swirls are unmistakeably feminine and invitingly sexual. In one of the most beautiful of the cases, Hosianna, the vegetable matter is arranged as a luxuriant pubic tangle; the prima mater from which life itself issues. Kiefer has managed somehow (perhaps by treating the case as a kind of bath) to run streams of graphite across the sand and then wash them out to form delicate rills that suggest the ferns and lichens of the first green life to appear on the living planet. Gustave Courbet's lavishly devotional hymn to the pudenda, The Origin of the World, is by contrast all frisky slickness. Instead of the erotic quiver, Kiefer gives us a heart-stopping moment, as we suddenly read those marks as simultaneously biological and cosmological, micro and macro; a vision of deltaic capillaries, the pulsing veins and branches of an estuary as seen from an orbiting camera, or up close, the fronds waving gently through a transparent wash of nourishing water.

This is as good, I think, as art ever gets: mystery and matter delivered in a rush of poetic illumination. That Kiefer's work happens to engage with almost everything that weighs upon us in our tortured age - the fate of the earth, the closeness of calamity, the desperate possibility of regeneration amid the charred and blasted ruins - and that it does so without the hobnailed tread of pedestrian polemics, is just one of the many marvels for which we have to thank, yet again, this most indefatigable of modern magi.

· Anselm Kiefer: Aperiatur Terra is at White Cube, London SW1, from January 26 to March 17. Details: 020-7930 5373